The structured $\varepsilon$-stability radius is introduced as a quantity to assess the robustness of transient bounds of solutions to linear differential equations under structured perturbations of the matrix. This applies to general linear structures such as complex or real matrices with a given sparsity pattern or with restricted range and corange, or special classes such as Toeplitz matrices. The notion conceptually combines unstructured and structured pseudospectra in a joint pseudospectrum, allowing for the use of resolvent bounds as with unstructured pseudospectra and for structured perturbations as with structured pseudospectra. We propose and study an algorithm for computing the structured $\varepsilon$-stability radius. This algorithm solves eigenvalue optimization problems via suitably discretized rank-1 matrix differential equations that originate from a gradient system. The proposed algorithm has essentially the same computational cost as the known rank-1 algorithms for computing unstructured and structured stability radii. Numerical experiments illustrate the behavior of the algorithm.

翻译：引入结构化$\varepsilon$-稳定性半径作为评估线性微分方程解在矩阵结构化扰动下瞬态界鲁棒性的度量。该方法适用于通用线性结构，例如具有给定稀疏模式或受限值域和余值域的复数或实数矩阵，以及托普利兹矩阵等特殊类别。该概念在概念上将非结构化和结构化伪谱统一为联合伪谱，从而允许使用如非结构化伪谱的预解式界以及如结构化伪谱的结构化扰动。我们提出并研究了一种计算结构化$\varepsilon$-稳定性半径的算法。该算法通过源自梯度系统的适当离散化秩1矩阵微分方程求解特征值优化问题。所提出算法在计算复杂度上与计算非结构化和结构化稳定性半径的已知秩1算法基本相当。数值实验展示了该算法的行为特性。